Conjecture: Is it true that the limiting diameter of a (strictly !) nested family of compact sets must be 0 ??

Question

Consider a family of compact subsets of $\mathbb{R}^n, C_1 \supset C_2 \supset C_3 \ldots$.

Also, and this is the important bit,

1) $C_j$ has empty interior for all $j \in \mathbb{N}$

2) The inclusions are strictly decreasing, that is $C_i \neq C_j \; \forall i \neq j$

Can we then prove that $\lim_{k \rightarrow \infty}\;\mathrm{diam}(C_k) = 0$ ?

Answer 1

Consider $\mathbb{R}^m$,$m> 1$,and $C_n = [0, \frac{n+1}{n}]\times \{0\}^{m-1}$. For one dimension we can take decreasing Cantor sets in $[0,1]$ that all contain $0$ and $1$ (so have diameter $1$).